Eric S. Rowland
The number of nonzero binomial coefficients modulo p^alpha.
Wed, 01/13/2010 - 16:01 — SomNambulAuthors: Eric S. Rowland
Subjects: Number Theory
AbstractIn 1947 Fine obtained an expression for the number a_p(n) of binomial
coefficients on row n of Pascal's triangle that are nonzero modulo p. One can
set up a recurrence for the number of integers 0 <= m <= n such that there are
b borrows involved in subtracting m from n in base p; Kummer's theorem renders
this recurrence as a generalization of Fine's theorem, giving a way to compute
the number a_{p^alpha}(n) of nonzero binomial coefficients modulo p^alpha.Iterated primitives of logarithmic powers.
Mon, 11/09/2009 - 16:01 — SomNambulAuthors: Eric S. Rowland, Luis A. Medina, Victor H. Moll
Subjects: Number Theory
AbstractThe evaluation of iterated primitives of powers of logarithms is expressed in
closed form. The expressions contain polynomials with coefficients given in
terms of the harmonic numbers and their generalizations. The logconcavity of
these polynomials is established.p-regularity of the p-adic valuation of the Fibonacci sequence.
Fri, 10/16/2009 - 17:25 — SomNambulAuthors: Eric S. Rowland, Luis A. Medina
Subjects: Number Theory
AbstractThis paper studies the $p$-adic valuation of the sequence $\{F_n\}_{n \geq
1}$ of Fibonacci numbers from the perspective of regular sequences. We
establish that this sequence is $p$-regular for every prime $p$ and give an
upper bound on the rank for primes such that Wall's question has an affirmative
answer. We also point out that for primes $p \equiv 1,4 \mod 5$ the $p$-adic
valuation of $F_n$ depends only on the $p$-adic valuation of $n$ and on the sum
modulo $p-1$ of the base-$p$ digits of $n$ -- not the digits themselves or
their order.p-regularity of the p-adic valuation of the Fibonacci sequence.
Fri, 10/16/2009 - 17:25 — SomNambulAuthors: Eric S. Rowland, Luis A. Medina
Subjects: Number Theory
AbstractThis paper studies the $p$-adic valuation of the sequence $\{F_n\}_{n \geq
1}$ of Fibonacci numbers from the perspective of regular sequences. We
establish that this sequence is $p$-regular for every prime $p$ and give an
upper bound on the rank for primes such that Wall's question has an affirmative
answer. We also point out that for primes $p \equiv 1,4 \mod 5$ the $p$-adic
valuation of $F_n$ depends only on the $p$-adic valuation of $n$ and on the sum
modulo $p-1$ of the base-$p$ digits of $n$ -- not the digits themselves or
their order.Growing words in the free group on two generators.
Fri, 09/04/2009 - 12:10 — SomNambulAuthors: Bobbe J. Cooper, Eric S. Rowland
Subjects: Group Theory
AbstractThis paper is concerned with minimal length representatives of equivalence
classes of F_2 under Aut F_2. We give a simple inequality characterizing words
of minimal length in their equivalence class. We consider an operation that
"grows" words from other words, increasing the length, and we study root words
-- minimal words that cannot be grown from other words. Root words are "as
minimal as possible" in the sense that their characterization is the boundary
case of the minimality inequality.
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